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Abstract

We demonstrate a general method for the first order compensation of singularity splitting in a vortex beam at a single plane. By superimposing multiple forked holograms on the SLM used to generate the vortex beam, we are able to compensate vortex splitting and generate beams with desired phase singularities of order ℓ = 0, 1, 2, and 3 in one plane. We then extend this method by application of a radial phase, in order to simultaneously compensate the observed vortex splitting at two planes (near and far field) for an ℓ = 2 beam.

(a)–(d) Far-field images of beams with OAM ℓ = 0, 1, 2 and 3 in the absence of any compensation. The configuration of lenses for these images was L3 and L4 (Fig. 2). (e)–(f) Near-field images of the same beams are given for ℓ = 0, 1, 2 and 3. The configuration used for these images was L3 and L5 (see Fig. 2). From these images (a)–(d) and (e)–(h), we can see that the effect of aberrations on a vortex beam is to induce a splitting of the central singularity and a deformation of the intensity profile. This is consistent with the existence of another mode with ℓ1 = ℓ − 2. In all images a logarithmic intensity scale is used. The physical scale used in all images is the same 490px×490px (3mm×3mm).

To construct the phase hologram used to generate two collinear beams with OAM ℓ = 2 and ℓ′ = 0, we calculate the complex amplitude of each individual beam, add them and extract the resulting phase profile (ψSLM). Parameters α′ and ϑ′ control the relative weighting and phase between the beams. When illuminated with a Gaussian beam, the output is a ℓ′ = 0 and a ℓ = 2 beam. Although not additive, displayed are the holograms associated with the ℓ′ = 0 component, ℓ = 2 component and the final hologram used to create the two collinear beams.

Far-field images of beams with OAM quantum numbers of ℓ = 0, 1, 2 and 3 (a)–(d) with compensation applied as detailed in the text. The deformation of the intensity profiles are decreased compared to Figs. 3(a) and 3(b). Vortex splitting associated with ℓ = 3 and 4 (c,d) has been reduced relative to Fig. 3. Near-field images of the same beams are given for ℓ = 0, 1, 2 and 3 (e)–(f). In all images a logarithmic scale is used.

Far-field (a) and near-field (b) images of a beam with OAM ℓ = 2 in the absence of any compensation. Far-field (c) and near-field (d) images of an ℓ = 2 beam compensated by the removal of the ℓ1 component by destructive interference with a collinear beam of ℓ′. Far-field (e) and near-field (f) images of a ℓ = 2 beam with the same compensation as applied in (c),(d) but with a radial phase (Eq. (10)) applied to the ℓ′ = 0 compensation component. For the radial phase, r0 = 10 pixels (90 μm). The result of this additional radial phase is that the separation of the vortices in the near-field has been reduced from (c) to (e). In all images a logarithmic scale is used.